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A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x. 6
0, 1, 3, 7, 16, 33, 66, 125, 231, 412, 720, 1227, 2056, 3380, 5478, 8745, 13792, 21483, 33114, 50510, 76344, 114356, 169920, 250503, 366666, 532975, 769758, 1104847, 1576640, 2237331, 3158208, 4435502, 6199479, 8624820, 11946096, 16475880, 22630864, 30962990 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = A273845(n)/3, n > 0.

G.f.: ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3.

a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

EXAMPLE

G.f. = x + 3*x^2 + 7*x^3 + 16*x^4 + 33*x^5 + 66*x^6 + ...

MATHEMATICA

nmax = 50; CoefficientList[Series[(Product[(1 - x^(3*j))/(1 - x^j)^3, {j, 1, nmax}] - 1)/3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

PROG

(PARI) first(n)=my(x='x); concat([0], Vec((prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))-1)/3)) \\ Charles R Greathouse IV, Nov 07 2016

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A)^3 - 1) / 3, n))}; /* Michael Somos, Nov 13 2016 */

CROSSREFS

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), this sequence (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Sequence in context: A219846 A229914 A192968 * A217942 A002936 A014668

Adjacent sequences:  A277965 A277966 A277967 * A277969 A277970 A277971

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Nov 07 2016

STATUS

approved

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Last modified June 19 06:47 EDT 2019. Contains 324218 sequences. (Running on oeis4.)